what does z|G mean in nonparametric bayes?

Question

when saying z is distributed according to G, where G comes from a dirichlet process, i saw this expression:

z|G ~ G

is this same meaning with z ~ G ?

Answer 1

It's not quite the same as $Z\sim G$.

• $z|G\sim G$ means that if you condition on the value of $G$, which is a distribution,$z$ comes from that distribution.
• $z\sim G$ means that $z$ comes from the distribution $G$.

To see the difference, consider a much simpler case: $X\sim N(0,\sigma^2)$ vs $X|\sigma^2\sim N(0,\sigma^2)$

• $X\sim N(0,\sigma^2)$ says $X$ has a Normal distribution (with mean $0$ and variance $\sigma^2$)
• $X|\sigma^2\sim N(0,\sigma^2)$ says $X$ has a Normal distribution conditional on $\sigma^2$ (with mean $0$ and variance $\sigma^2$), so $X$ has a Normal scale-mixture distribution unconditionally. If, for example, $\sigma^2$ had a $\chi^2_1$ distribution, $X$ would have a $t_1$ distribution unconditionally